COMPOSITIO MATHEMATICA

YASMINE B. SANDERSONDimensions of Demazure modules for ranktwo affine Lie algebrasCompositio Mathematica, tome 101, no 2 (1996), p. 115-131<http://www.numdam.org/item?id=CM_1996__101_2_115_0>

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Dimensions of Demazure modules for rank

two affine Lie algebras

YASMINE B. SANDERSONInstitut de Recherche Mathématique Avancée, Université Louis Pasteur et C.N.R.S.,7, rue René-Descartes, 67084 Strasbourg Cedex, France.

Received 2 August 1994; accepted in final form 6 February 1995

Abstract. Using the path model for highest weight representations of Kac-Moody algebras, wecalculate the dimension of all Demazure modules of A(1)1 and A(2)2.

Compositio Mathematica 101: 115-131, 1996.© 1996 Kluwer Academic Publishers. Printed in the Netherlands.

1. Introduction

Let g be a symmetrizable Kac-Moody algebra. Recall that for every dominantweight À, there exists a unique (up to isomorphism) irreducible highest weightmodule V = V(03BB) of highest weight a . In order to gain some insight into thestructures of these modules, one can study their characters. Recall that these areformal sums ~(V) = £(dim Vtt)ett over all weights J1 and where Vtt is the weightspace of V of weight J1. Another reason to study character formulas is because oftheir connection with combinatorial identities. For example, the Rogers-Ramanujanand Euler identities have been derived from specializing the characters of certainlevel 2 and level 3 highest weight modules for A(1)1 (see [LW]).

For symmetrizable Kac-Moody algebras, character formulas for all highestweight modules exist, the most well-known being Weyl’s formula, due to Kac[Ka]. Unfortunately, none of these expressions are combinatorial (they containmany + and - signs) making it difficult to obtain from them a general formula fordim V, for an arbitrary weight J1.

There is an analogous problem for Demazure modules. Recall that for everyelement w of the Weyl group, the weight space Vw(03BB) is of dimension one. Let Ew(03BB)denote the b-module generated by VW(À)’ where b is the Borel subalgebra of g. Thefinite-dimensional vector spaces Ew(03BB) are called Demazure modules and form afiltration of V. These modules often occur in inductive proofs and are connectedwith the Schubert varieties of the Kac-Moody algebra. Character formulas alsoexist for Demazure modules [D], [Ku], [M], but they are not combinatorial. Sofar, very little is known about them. In particular, their dimensions have not beendetermined except in isolated cases.

In this paper, we determine explicit closed form expressions for the dimensionsof all Demazure modules for A(1)1 and A(2)2. Let ao and a 1 (respectively av, af)

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be the two simple roots (resp. coroots). Let Ao and 039B1 be the two fundamental

weights defined by ~039Bi, 03B1j = hij. In the case g = A(2)2, we choose the rxi suchthat ~03B10, 03B11~ = -1 and ~03B11, 03B10~ = -4. Let ri be the reflection with respect toai. The Weyl group W is generated by the ri . For every n &#x3E; 0, W contains twoelements of length n

THEOREM 1. Let A - sAo + tAl.(1.1) If g - A(’), then:

(1.2) If g = A(2)2, then:

This result was known previously for the case g = A(1)1 when À = Ao or 039B1,[LS]. The main tool used in establishing these formulas is Littelmann’s path modelfor highest weight representations [L1]. The path model is a recently discoveredcombinatorial parametrization for a base for integrable modules. In Littelmann’swork, the base is parametrized by certain piecewise linear paths whose images lie in~*, where ~ is the Cartan subalgebra of our Kac-Moody algebra. A brief summaryof this theory is given in the Section 3. In this paper, we start off with a specific typeof path, called Lakshmibai-Seshadri (or L-S) paths, defined in [L1]. By workingfrom the definition of L-S paths, we obtain an explicit description of all such pathsfor the basic modules. This approach, however, is limited to the basic modules.When À is other than a fundamental weight, the calculations needed to explicitlydescribe L-S paths quickly become unwieldy, if not impossible. The idea that weuse to bypass this problem is the following: for any highest weight A = sllo + t039B1,start off with the path

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The path 03C0 is a concatenation of L-S paths representing the fundamental weights.We then obtain an explicit description of the set of paths in the associated pathmodel. These paths will also be concatenations of L-S paths. This description willthen allow us to describe dim Ew(03BB) as an induction on w and A. In the case ofA(1)1, the parametrization obtained with this method coincides with that given bystandard monomial theory [LS]. Although the proofs for these explicit descriptionsare only given for the rank two affine cases, they are not dependent on the algebrasbeing affine and it is easy to see that they would generalize to the case of higherranks.

2. Preliminaries

We recall a few basic facts about Kac-Moody algebras. We follow the notationsin [Ka]. Let g be a complex symmetrizable Kac-Moody algebra of rank n. As analgebra, g is generated by ei, b, and f for 0 i n - 1 where b is the Cartansubalgebra of g. Let b be the Borel subalgebra of g. We have that h ~ ~n-1i=0C03B1iand that b is the subalgebra generated by h and the ei. For 0 i n - 1, wedenote the simple roots by ai . The ai are the elements of h* defined by the relation[ h, e 2 ] = ~03B1i, h~ei for all h ~ h . In addition, for 0 i z n - 1, we denote thefundamental weights by 039Bi. These are elements of h* defined by the relations

(Ai, a’f) = 03B4ij. For all i, let r2 be the reflection of h* with respect to a,. Bydefinition, the Weyl group W is generated by the ri .

Let P+ = {03BB ~ ~ *| ~03BB, 03B1i~ ~ Z0} denote the set of all dominant weights. Forevery A e P+, there exists a unique (up to isomorphism) highest weight moduleV = V(03BB) of highest weight À. By definition, J1 e h* is called a weight if thecorresponding weight space V03BC = {v e V 1 h - v - (J1, h~v V h ~ h} is non-zero.We have V = E9 Vtt where the sum is over all possible weights. For every w e W,the weight space Vw( A) is of dimension 1.

Let Ew(03BB) be the b-module generated by VW(À). The Ew(03BB) are called Demazuremodules. They are finite-dimensional subspaces of V that satisfy the followingproperty: for any w, w’ e W such that w z w’ (where is the Bruhat order onW), we have Ew(03BB) g Ew’(03BB). In addition, UwEW Ew(03BB) = V.

3. The path model

The path model, introduced in [Ll], [L2], gives a combinatorial parametrizationof the basis vectors of a highest weight module for a Kac-Moody algebra. Thisparametrization is in the form of certain piecewise linear paths 7r : [0,1] - h* . Wenow give a brief synopsis of those parts of the theory necessary for this paper. (See[L1], [L2] for more details).

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3.1. GENERAL THEORY

Let 111 denote the set of all continuous piecewise linear paths 7r : [0, 1] ~ h* with7r(0) = 0 and 03C0(1) a weight. We identify any two such paths if their imagescoincide. To every simple root cxi, we associate linear operators ei, fi : 03A0 ~ II.The e2, fi, which we call root operators, act in the following way. Let T E 11 bea path. According to the behavior of the function t ~ ~(t), 03B1i~, either fiT isundefined or f i T is a new path in 11 such that fi(1) = (1) - ai. Likewise, eitherei T is undefined or ei T is in II with ei(1) = (1) + ai. Before we give a moredetailed description of their action, we will need the following definitions.

For any two paths Tl , T2 e 11, we denote by Tl * 2 their concatenation,

For any path T e lI and any ri e W, we define the path (ri)(t) := ri((t)). Letmi() := mint~[0,1]~(t), ay).We can now describe the action of fi. Let p E [0, 1] be maximal such that

(r(p), 03B1i~ == mi( r). If ~(1), 03B1i~ - mi(T) 1, then fiT is undefined. If not, wedo the following. Let x E [p, 1] be minimal such that ~(x), ay = mi() + 1. Wenow ’eut’ r into three parts Ti, T2 and T3. Each of these parts is a path in 111 and

they are defined by:

Notice that T = 1* T2 * T3. We define fi : = Tl * ri T2 * T3.We define ei similarly. Let q e [0,1] be minimal such that (T( q), 03B1i~ = mi().

If - mi() 1, then ei T is undefined. If not, we do the following. Let y E [0, q]be maximal such that (T(y), 03B1i~ = mi(T) + 1. We now ’eut’ T into three parts ri,T2 and T3. Each of these parts is a path in II and they are defined by:

Notice that T = Tl * T2 * T3. We define ei = Tl * TiT2 * T3.These definitions of the root operators are those found in [L1]. A second, more

general and slightly more complicated, definition is given in [L2]. This seconddefinition allows the path model to include a wider class of paths. However, forthose paths that we will be using (concatenations of Lakshmibai-Seshadri paths,defined below) these two definitions coincide. Therefore, we only give the simplerversion.

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With these two root operators we can define an operator Di on Z[II], the freeZ-module with basis 11

Choose a path 03C0 E II whose image lies in the fundamental chamber and whoseendpoint 03C0(1) is a dominant weight À. Such a path, 1r, is called a dominant path.Let B03C0 = {fi1 ··· fik Jr 1 ij e [0,..., n-1], k 01 be the set of all possible pathsthat one obtains by applying the root operators to 7r. We call B03C0 the path modelassociated to 03C0. We then have:

THEOREM 2. [L1]. X (V) LTEB1r eT(I).We also have an analogue for Demazure modules. Let w = Tjl... Tjk be

a reduced decomposition of w. Let Pw03C0 = Dj1 ··· Dij03C0 = {fi1j1···fikjk03C0|il, ... , i k C Z0}. Then

THEOREM 3. [L1]. ~(Ew(03BB)) = LTEPw1r eT(1)-From this description, we see that, once we have chosen a dominant path 1r, the

problem is to characterize all of the paths contained in B03C0 (or Pw03C0). In general,this is not an easy task. However, for one particular type of dominant path, thishas already been done for us [L1]. The dominant path in this case is the straightpath 03C003BB(t) = t À from 0 to À. The paths in the associated path model are calledLakshmibai-Seshadri paths.

3.2. LAKSHMIBAI-SESHADRI PATHS

For a given A E P+, we choose as dominant path 1r,B(t) = tA. The elements inB03C003BB are called Lakshmibai-Seshadri (or L-S) paths of shape À. These paths can becharacterized as follows. There is a bijection between paths T E B1r,B and pairs ofsequences (03C3, a) that satisfy the following conditions:

where Wa is the stabilizer of À in W and where &#x3E; is the relative Bruhat order. In

addition, we require that for every i ~ [1, ... , n - 1 ], there exists an ai -chain forthe pair (ai , 03C3i+1). Recall that, by definition [L1], the existence of an a-chain fora pair (03C3, eT’) of cosets in W/W03BB means that there exists a sequence of cosets inW/Wa

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where the r03B2i are the reflections with respect to the positive real root (3i, such thatfor all i E [1,... , s ]

We shall also refer to such a pair (e, g) as Lakshmibai-Seshadri (or L-S). Forany such pair, the corresponding path is

Finally, we will need

THEOREM 4. [L1].

4. Description of L-S paths for the basic modules

From here on, we assume that g is isomorphic to A(1)1 or to A(2)2. In this section,we will work directly from the definition of L-S pairs in order to obtain an explicitcharacterization of all such pairs. Note that

On each of these two coset spaces, the Bruhat order is a total ordering. For03B5 ~ {- , +} , w03B5n &#x3E; w03B5m ~ n &#x3E; m. For n 0, set:

In the case of A(1)1, ~03B10, 03B11~ = ~03B11, 03B10~ = -2. In the case of A(2)2, to fixnotation, we choose the ai such that (ao, 03B11~ = -1 and ~03B11, 03B10~ = -4.

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LEMMA 1. Let 039B be a fundamental weight. The L-S paths 7r of shape 039B are thosepaths 7r = (Q, a) such that

wheree = +if A= 039B0, 03B5 = if 039B = 039B1 and where a03B5j · d03B5j E /Z.Proof. Clearly, 7r is a path of shape A. We need to show that the chain condition

is satisfied by these paths only. Notice that dj and d03B5j+1 are relatively prime. There-fore, if a e Q, 0 a 1 is such that a - d03B5j E z, then a - d03B5j+1 ~ Z. Therefore,cannot be L-S unless a is of the form above. In other words, there can be no ’skips’in the sequence of Weyl group elements. The chain condition demands that a03B5j beas stated above. D

Lemma 1 shows us that for any given 03C3, there exists possibly more than one a suchthat 7r == (03C3, a) is L-S. For any L-S path 1r = (ff, a) where 01 : 03C31 &#x3E; ... &#x3E; 03C3rdefine beg(03C0) := ul and end(03C0) := cr r. For A one of the fundamental weights,let

For ease of notation and when there can be no ambiguity conceming which funda-mental weight is A, we will write p( m, n ) instead of p039B(m, n). In addition we willwrite Pn (03C0s039B0) for P + (1r SAO) and Pn(03C0s039B1) for P - (03C0s039B1).

In order to determine the dimensions of any given Demazure module of g, wewill first calculate 1 p,,, (m, n) and |p039B1 (m, n) for m n 0. This we do for AP)and A(2)2 in Sections 4.1- 4.2. Notice that these results will immediately give thedimension of any Demazure module for Ao and 039B1 because

and similarly for dim Ewn (A 1).

4.1. CHARACTERIZATION OF L-S PATHS FOR THE BASIC MODULES OF A(1)1Because of the symmetric roles of ao and 03B11, we need only consider L-S paths ofshape Ao. The lemma above states that 7r = (03C3, a) is in p039B0(m, n) if and only if

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where i j ~ {1, ... , j-1} for n+1 j m.A(m-n)-tuple(im,im-1...,in+1) ENk satisfies these inequalities if and only if 1 im im-1 ··· in+1 n.There are (m-1n-1) such sequences. We conclude that for all m n &#x3E; 0, lp(m, n) | =(m-1 n-1). Moreover lp(o, 0) == 1 and lp(m, 0) | = 0 for m &#x3E; 0. Thus it will be con-

venient to set (-1 -1) = 1 and (m-1 -1) = 0 for m &#x3E; 0.

PROPOSITION 5. Let g = A(1)1 and A be a fundamental weight. Then

dim Ew(03BB) = 2n where n is the length of w E W /WA.Proof Without loss of generality, we can assume that A = Ao. Then Pn (1r AO) =

03A30i,jn p(i,j). Therefore, dim Ew(039B0) = |Pn(03C0039B0)| = 03A30jin |p(i,j)| =

This result was obtained previously by V. Lakshmibai and C.S. Seshadri, (see[LS], (2) p. 194 ).

4.2. CHARACTERIZATION OF L-S PATHS FOR THE BASIC MODULES OF A(2)2We first consider L-S paths of shape Ao. Set a(m, n) := |p039B0(m, n)|.LEMMA 2. The a(m, n) satisfy the following recursion relations:

Proof. Clearly, a(m, m) = 1. For m &#x3E; n 0, let (e, 1) E PAo(m,n). ByLemma 1, a is of the form

where Z*, is even whenever j is even. Then (Ím, ..., in+1) E Nm-n satisfies theseinequalities if and only if

Setting kr :=ir + m - r, we see that this is equivalent to

where, for j even, k, = m mod 2.If n is even, we claim

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To obtain this identity, for each i e {1, ... , n/21, fix kn+2 = m - 2i. Then thenumber of possible subsequences (km, ... kn+3) that satisfy 1 km ... kn+4 kn+3 kn+2 - 1 (with the same parity requirements) equals a(m -2i, n - 2i + 2). There are still 2i - 1 possibilities for kn+ 1. Now sum over i. If nis odd,

To see this identity, for each i ~ {1,..., (n - 1)/2}, fix kn+1 1 = m - 2i. Thenthe number of possible subsequences (km,..., kn+2) that satisfy 1 km ... kn+3 kn+2 kn+1 - 1 (with the same parity requirements) equalsa(m - 2i, n - 2i + 2). Now sum over i. D

REMARK. The a(m, n ) can be calculated explicitly. For k and j such that m &#x3E;

n 0 below,

where (x y) = 0 whenever x y or y 0.

We now consider L-S paths of shape AI. Set b(m, n) := IPAI (m, n) 1.LEMMA 3. The b(m, n) satisfy the following recursion relations:

R3. b(m, 2k) = b(m - 2,2k) + 2b(m - 2,2k - 1) + b(m - 2,2k - 2)

R4. b(m, 2k + 1) - 3b(m - 2,2k + 1) + 4b(m - 2,2k) + b(m - 2,2k - 1)

Proof. By Lemma 1, b(m, n) is the cardinality of the set of (m - n)-tuples(im, ... , in+1) E Nm-n such that

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and where ij is even if j is odd and n + 1 j m.To see what conditions the ( im, ... , in+1) must satisfy we do the following. For

each j e {n, ... , m} look at the subset of these (m - n)-tuples that satisfy

This is true if and only if

Setting kr := iT + m - r for j + 1 r m and kr := iT + m + j - 2r forj r n + 1, we see that this is equivalent to

where for r odd, r j + 1, kr = m + 1 mod 2 and for r odd, r j , kr = j + mmod 2. Note that 03A3mj=n bj(m, n) = b(m, n). We now determine bj(m, n).

The number of ( m - j )-tuples that satisfy

(with the appropriate parity conditions on kr for r odd) is a(m + 1, j + 1 ). Thenumber of ( j - n)-tuples that satisfy

(with the appropriate parity conditions on kr for r odd) equals a(j + 1, n + 1) if jis odd. If j is even, then it equals

Then bj(m, n) is just the product of these two cardinalities. The recursion rela-tions are obtained by applying R1-R2 to each summand of b(m, n) =¿T=nbj(m,n). 0

5. Description of paths for all other modules

When our dominant weight A is not one of the fundamental weights, it is difficultto calculate the dimensions of the Demazure modules directly by counting the

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number of L-S paths of shape A. Such a path 7r = (g, a) can be one where g has’skips’ in its sequence of Weyl group elements. For example, 0, could resemblew03B5m &#x3E; w03B5m-2 &#x3E; w03B5m-3 &#x3E; ... &#x3E; w-. In addition, if A is regular, E is variable.The difficulty with keeping track of such g makes counting such L-S paths a realpain.

In order to bypass this problem, we choose a dominant path that is a concatena-tion of L-S paths for the basic modules. Suppose that À = sAo + t039B1. Set

This path traces a straight path from 7r(0) = 0 to sAo and then one from sAo to7r( 1 ) = À. Note that this is an L-S path if and only if st = 0. For two sets Sl, S2 ofpaths, let S1* 52 := {03C01 * 03C02 |03C01 C Sl, 7r2 C S2}. There is a natural injection

The problem then is to determine exactly which paths lie in the image of B03C0.

THEOREM 5. Let 03BB = s039B0 + t039B1 ~ 0. Let 03C01, ... , 7r S E B03C0039B0 and let (1, ... , (t EB 7r Al. Then 7r 1 * ... *03C0s * (1 * ... *03B6t E Pw(03C0s039B0 * 03C0t039B1) if and only if

C1. end(03C0i) beg(03C0i+1) for all i and end (03B6j) beg( (j+1) for all j.

C2. end(03C0s) · r1 beg(03B61)

C3. If s = 0, then beg(03B61) w. If s &#x3E; 0, then beg(03C01) w. In addition, ifend(03C0s) beg( (1), then beg(7rl) - riz w.

Proof. The proof is as follows:

(1) We first show that for 7r1, ... , 7r S E B03C0039B0, we have 03C01* ··· * 7r, C B7r,,,,if and only if the 03C0i satisfy Cl. (In fact, the same proof shows that for03B61, ... , (t E B7r Al’ we have 03B61* ···*03B6t e B7rt,,, if and only if the (i satisfyCl.)

(2) We then show that for 7r e B1r SAO and for ( e B03C0t039B1, if 03C0 * 03B6 ~ B(03C0s039B0 * 03C0t039B1)then 7r * ( must satisfy C2.

(3) Finally, we show that for 7r E B03C0s039B0 and ( e B03C0t039B1, we have 03C0 * 03B6 EPw(03C0s039B0 * 7r tA l) if and only if 7r * ( satisfies C1-C3. This proof is by inductionon w and implies the ’only if’ direction for 2.

Proof of 1.: Any path in B(03C0s039B0) is obviously the concatenation of s paths thatsatisfy condition C 1. We now show that the concatenation of any s paths 03C01, ... , 7r Sthat satisfy Cl is in B(03C0s039B0), We proceed by induction and suppose that we have

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shown this for s - 1 paths. Let 7r, E B1r(s-l)AO and let 7r2 E B(03C0039B0). Being L-Spaths, 03C01 = (03C3, a) and Jr2 = (T, b) are defined as follows:

If end(03C01) = beg(03C02), (in other words, if cr r = Tl) then define 7r = (03BA, c) where

where ai = ai(s-1) s for 1 i r and ai-1 = bi+s-1 s for r + 1 i r + q.If end(03C01) &#x3E; beg(03C02), (in other words, if ar &#x3E; Tl ) then define 7r - (k, c)

where

where ci = at(:-1) for 1 i r and c,-i 1 = b,+,-l for r i r + q.7r = (k, c) is obviously a path of shape sllo. Again, to show that it is

L-S, we need only check the chain condition. This is immediate for 7r becauseif w e WjWAo and a, a positive real root, are such that ai~w(039B0), aV) ~ Z (resp.bi~w(039B0), aV) E Z) then

Proof of 2. : We now show that if 7r E B03C0s039B0 and ( E B1rtAI are such that03C0 * 03B6 e B(03C0s039B0 * 1rtAl) then property C2 must hold. In fact, we need only showthat the root operators preserve property C2; since the dominant path 03C0s039B0 * 7r tA 1satisfies C2 then so should all paths in B(03C0s039B0 * 03C0t039B1).

For any two paths r, o- recall that

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If ei(03C0*03B6) == 03C0*ei03B6 then beg(ei03B6) = beg(03B6) or beg (ei03B6) = ri·beg(03B6) beg(()so ei(03C0 * () satisfies C2. Suppose now that ei( 7r * () = ei7r * (. If end( ei7r) ==end(03C0) then ej (Jr * () satisfies C2. Suppose then that end( ei7r) = r, - end(03C0) end(03C0). This would happen when mi(03C0 * () occurs at the endpoint of 7r. Then~end(03C0)(039B0),03B1i~ 0 and ~beg(03B6)(039B1),03B1i~ &#x3E; 0 so r, . end(03C0)r0 beg(03B6) andei(r * () satisfies C2. A similar argument shows that the action of the fi preservesproperty C2.

Proof of 3. : If s = 0, then 1r Al * ··· 7rA 1 - 1rtAI is a L-S path of shape t039B1.Therefore, C3 follows immediately from Cl and Theorem 4. Now assume thats &#x3E; 0. Let w E W. We now show that a path 7r * ( lies in Pw(03C0s039B0 * 03C0t039B1) if andonly if beg(03C0) w and, if end(03C0) beg(03B6), then beg(03C0) · r1 w.

(i) Set w = riu where u w. We will assume that Pu :== Pu (1r SAD * 1rtAI) equalsthe set of paths that satisfy C1-C3.

(ii) Set P£ equal to the set of paths in B(03C0s039B0) * B(03C0t039B1) that satisfy C1-C3. Wewill be done once we have shown that P’w = Us0 fsiPu. This is equivalentto showing

We show (a). Let 03C0 * 03B6 E P’wBPu. Let v = beg(7r). If w equals some w+n,then v = w+n. If w equals some w-n, then v = w+n-1. In any case, we have~v(03BB),03B1i~ -1.So, there exists some power l &#x3E; 0 such that beg(eli03C0) = r2w = u.Therefore, there exists a power n 1 such that eni(03C0 * () = eli(03C0) * en-li(03B6) ~ O.Conditions C1-C2 force beg(eli(03C0)) · r1 u if end(eli(03C0)) beg(cn-li(03B6)). There-fore, ei (1r * () E Pu. The proof of (b) is similar. 0

In the case of A(1)1, this parametrization of the basis vectors is equivalent tothat given by standard monomial theory [LS]. Theorem 5 implies the followingrelations, which will be used to prove Theorem 1. Here, let A denote any of thefundamental weights.

The last relation comes from using an analogue of Theorem 5 for the path

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This path traces a straight path from 7r’(0) - 0 to tA, and then one from tA, to03C0’(1) = A.

6. Proof of Theorem 1

Proof for g = A(1)1. We first prove our formula for the case À fl 0 is singular(i.e. st = 0). We have already proved the formula for A a fundamental weight. ByTheorem 3 and relation El,

and similarly for dimE -(tA1). Now suppose that A - sllo + t039B1 regular. Then,by Theorems 3, 5 and relation E2,

Use relation E3 to obtain dim E + (A).

Proof for 9 = A(2)2. As in the case above, we first prove the dimension formulasfor A singular and then for A regular.

Case A = sAo. We must show that dimE + (sAo) = 2-n(s + l)n(s + 2)n andthat dim E + (sAo) = 2-n(s + 1)n+1(s + 2)n.w2n+ I

The proof is by induction on n. By using the definition of L-S paths, wecan compute the following directly for all s: dim Ew+1 (s039B0) = (s + 1) and

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dim Ew+2 (s039B0) = !( s + 1)(s + 2). Set Rn := Pn(03C0s039B0)BPn-1(03C0s039B0). We needto show that |Rn| = 1 2(s + 1)(s + 2) |Rn-2| for n &#x3E; 2. By relation El, Rn =

03A3j0 p(n,j) * Pj (7r (s-l )AO). Therefore,

Equation 1 was obtained by using RI and R2. Equation 2 was obtained by reorderingand matching indices.

Case A = t039B1. We must show that dim Ew-2n (t039B1) = (t + 1)n(2t + 1)" and thatdim E - (IAI) = (t + 1)n+1(2t + 1 )n.

The proof is by induction on n. By using the definition of L-S paths, wecan compute the following directly for all t : dim Ew+1(t039B1) = (t + 1) and

dim Ew+2(t039B1) = (t + 1)(2t + 1). Set Rn := Pn(03C0t039B1)/Pn-1(03C0t039B1). We needto show that IRnl - (t + 1)(2t + 1) |Rn-2| for n &#x3E; 2. By relation El, Rn =

03A3j0 p(n,j) * Pj(7r(t-1)Al). Therefore,

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Equation 3 was obtained by using R3 and R4. Equation 4 was obtained by reorderingand matching indices.

Case A = sAo + t039B1 regular:The proof is by induction on s and t. For l(w) 2 one can verify the dimensions

formulas for all s and t by hand using Theorem 5 and relations E2-E3. Let n &#x3E; 2.

Set Rw : Pw-(03C0s039B0 *03C0t039B1)/Pw - (7r SAD *03C0t039B1). Then, by Theorem 5 and relation- Wn n-1E2,

Equation 5 was obtained just as in the cases above for À singular, that is, by applyingR1-R2 and then rearranging indices. Therefore, the dimensions hold for all Wn -

Now, let Rw+n : = Pw+n(03C0t039B1 * 03C0s039B0)BPw+n-1(03C0t039B1 * 03C0s039B0). By relation E3,

Equation 6 was obtained just as in the case above when À is singular, that is, byapplying R3-R4 and then rearranging indices. Therefore, the dimensions hold forall w+

As we can see from Theorem 1, for fixed w e W, dim Ew (sAo + t039B1) arepolynomials in the variables s and t. This suggests that each basis vector can beparametrized by an integral point in a certain lattice polytope of dimension equalto the length of w. In the case of the basic modules, this is clearly the case. Ifn equals the length of w, then each path is represented by a lattice point in thesimplex with vertices vo, v1, ··· , vn where vi = (d03B51, d2, ..., di , 0, ... , 0). Thendim Ew(s039B0) (respectively dim Ew(t039B1)) equals the number of lattice points inthat simplex dilated by a factor of s, (resp. t). Raika Dehy (Rutgers University,

131

personal communication) has established the existence of a corresponding polytopefor the regular highest weights.

Acknowledgements

The author would like to thank F. Knop, P. Littelmann and O. Mathieu for usefulconversations.

References

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